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\centerline{\bf Surfaces in $\P^3$ over finite fields }
\medskip
\centerline{\bf Jos\'e Felipe Voloch}

\bs
Abstract: For a smooth surface $X$ in $\P^3$ of degree $d$,
defined over a finite field $\F_q$ with $q$ elements, $q$ prime,
we prove that $X$ has at most
$d(d+q-1)(d+2q-2)/6 + d(11d-24)(q+1)$ points with coordinates in $\F_q$.
\bs

{\bf 0. Introduction}
\bs

In this paper we will obtain an upper bound on the number of rational
points on a surface $X$ in $\P^3$ of degree $d$ over a finite field $\F_q$,
which in a certain range of $d$ and $q$ improves on all other known bounds.

The bound will be obtained by counting the number of points $P$ of $X$ in
an algebraic closure of $\F_q$ whose image under the Frobenius map lies
in one of the asymptotic lines to $X$ at $P$. 
Recall that the asymptotic lines
to a surface $X$ in $\P^3$ are the lines that touch $X$ at a point with 
multiplicity at least three. Usually, given a general point $P$ of $X$, there
are two asymptotic lines tangent to $X$ at $P$. We will
make this statement more precise later when we investigate the geometry of
asymptotic lines in detail. This is, of course, a very classical topic in
algebraic geometry but has only been investigated in characteristic zero.
We will obtain some new geometric results in positive characteristic and raise
a few questions.

This bound is the analogue for surfaces of theorem 0.1 of [SV], which is 
about plane curves. In [SV] a general result is proved
for morphisms of curves to projective spaces of arbitrary dimension with many
consequences and one would hope to similarly extend the results here. This seems
to be considerably harder but we hope to return to it in a later paper.

We will also compare our bound with the other known bounds for surfaces 
over finite fields and discuss some examples.


\bs

{\bf 1. Geometry}

\bs

In this section we fix an algebraically closed field $k$ of characteristic $p>0$.
Let $X$ be a smooth surface of degree $d>1$ over $k$ defined by the equation $f=0$, where
$f$ is a homogeneous polynomial of degree $d$ in $k[x_0,x_1,x_2,x_3]$.
We will study various geomtric properties of $X$.

We begin with the dual variety of $X$. Let ${\P^3}^{*}$ be the dual projective
space and $\gamma: X \to {\P^3}^{*}, P \mapsto T_PX$ be the Gauss map.
As usual, $T_P X$ denotes the tangent plane to $X$ at $P$, given by the equation
$ \sum_{i=0}^3 f_{x_i} (P) \, x_i = 0 $, where, here and
elsewhere, $f_{x_i}$ denotes the partial derivative $\partial f / \partial x_i$.
So $\gamma$ is given by 
$P \mapsto (f_{x_0}(P):\!f_{x_1}(P):\!f_{x_2}(P):\!f_{x_3}(P))$.
We denote $\gamma (X)$ by $X^{*}$, which is called the dual variety of $X$.
If $p$ does not divide $d(d-1)$, $\gamma$ is birational and $\deg X^{*} = d(d-1)^2$
(See, e.g. [HK]).

The points of ramification of the Gauss map are known as parabolic points and
the points where the derivative of the Gauss map is zero are known as planar points.
Every line through a planar point in the
tangent plane at the point touches the surface with multiplicity at least three.
On a non-planar point, the general line through the point in the tangent plane
at the point touches the surface with multiplicity two and by a local argument
it easy to see that, in such a point, there are either two or one lines touching
the surface with multiplicity at least three; these are the asymptotic lines. 
The parabolic points are exactly those
for which there is exactly one such line. We call the points for which there 
are exactly two such lines, ordinary points.
To obtain the asymptotic lines through $P\in X$ one considers the first and second polars 
to $X$ at a point $Q=(y_0:\!y_1:\!y_2:\!y_3)$, that is
$\sum f_{x_i} y_i, \sum f_{x_i x_j} y_iy_j$. The line $\overline{PQ}$
is an asymptotic line to $X$ at $P$ if and only if the first and second polars
to $X$ at $Q$ vanish at $P$.

If the Gauss map is inseparable, all points are parabolic (or planar). 
Otherwise the set of parabolic points form a curve, the parabolic curve with equation
$H= \det (f_{x_i x_j}) =0$ (the polynomial $H$
is the Hessian of $f$) and is therefore a curve of degree $4d(d-2)$.
Of course, the parabolic curve may not be reduced. For instance, planar points
are easily seen to be singularities of the parabolic curve and, therefore, if
the surface has a one-dimensional set of planar points, these will
form a multiple component of the parabolic curve.

We will call a surface ordinary if the Gauss map is birational and, therefore,
there is an open set of ordinary points. Let $\G$ be the Grassmannian of lines
in $\P^3$ and define $Z' = \{(P,L) \in X\times\G \mid (X \cdot L)_P \ge 3\}$, in
other words the set of pairs $(P,L)$ where $L$ is an asymptotic line at $P$.
If $X$ has a one-dimensional set of planar points, then $Z'$ has a component which
is a $\P^1$-bundle over this set, since every line in the tangent plane of a planar
point is asymptotic. We let $Z_0$ be this component of $Z'$ and define $Z$ to
be the union of the remaining components of $Z'$.
We call $Z$ the asymptotic double cover of $X$ and, indeed if $X$ is ordinary
$Z$ is a double cover of $X$. (The asymptotic double cover was introduced in [McS]
for surfaces without planar points).
We note that $Z$ does not have to be irreducible. We say that the surface $X$ is
non-split if its asymptotic double cover is irreducible and split otherwise.

A point $P \in X$ is called flecnodal and a line $L$ a flecnodal line through $P$
if $(X \cdot L)_P \ge 4$. The name flecnodal is the traditional one ([Sa], pg.277)
and evokes the fact that, if $P$ is an ordinary flecnodal point, the curve
$X \cap T_PX$ has a node at $P$ with one branch being inflectional at $P$, with
the line $L$ being the tangent to said branch. Modern authors ([K], [McS], [V])
refer to asymptotic flex points instead of flecnodal points, 
because the flecnodal points are the inflection points
of the asymptotic curves. Since the asymptotic curves are not algebraic and may
not exist in positive characteristic we prefer to use the term flecnodal.
It may happen, in positive characteristic, that all points on $X$ are flecnodal
and we conjecture that this is only possible when $p$ divides $d(d-1)(d-2)$. We will
prove some partial results towards this conjecture below.
It is an old result of Salmon (see [Sa], pg. 278) that the flecnodal points of $X$
are cut out by a homogeneous polynomial of degree $11d-24$ and the proof is valid
in arbitrary characteristic (see also [K], [McS], [V], 12.8).
Thus, if not all points are flecnodal, the set of flecnodal points form a curve,
the flecnodal curve, of degree $d(11d-24)$.

Define a rational map $\phi: Z' \to \P^2$ as follows. 
Fix a plane $H \subset \P^3$ and if $(P,L) \in Z', L \not\subset H$, define
$\phi(P,L) = L \cap H$. We claim that $\deg \phi = d(d-1)(d-2)$. Indeed, for
a general $Q$ in $H$, to find the inverse image of $Q$ we need to find the
points $P$ which annihilate the equation $f$ of $X$ and also the 
the first and second polars to $X$ at $Q$, these are equations of degree
respectively $d$,$d-1$ and $d-2$ and for all $Q$ the system will have
solutions, so $\phi$ is surjective. A surjective map between varieties of the
same dimension is generically finite so the system has finitely many solutions
for general $Q$. It follows that $\deg \phi = d(d-1)(d-2)$.
We will use $\phi$ extensively below.

We can also define another map $\phi': Z' \to \P^2$ as follows.
Take a generic point $P_0$ of $\P^3$
and consider the $\P^2$ of lines through $P_0$. Given an element $(P,L) \in Z'$
define $\phi'(P,L)$ as the plane spanned by $L$ and $P_0$, provided $P_0$ is not in
$L$. To compute the degree of $\phi'$ take a plane $H$ through $P_0$. The points in
$Z'$ with image $H$ are pairs $(P,L)$ with $L \subset H$, as it is easy to see.
To count these points, notice that the condition that $L \subset H$ and 
$(X\cdot L)_P \ge 3$ is the same as $L$ having triple contact with $X \cap H$ at $P$,
thus we are counting the number of inflections of the plane curve $X \cap H$ of
degree $d$ and they number $3d(d-2)$, therefore $\deg \phi' = 3d(d-2)$.
The computation of these degrees is presumably the meaning of eqs. 3 and 4 on 
pg. 236 of [Sc]. We will not use $\phi'$ in what follows.

\proclaim Theorem 1. Not all points are flecnodal on a non-split smooth surface of 
degree $d$ if $p$ does not divide $d(d-1)(d-2)$.

{\it Proof:} Note that the hypotheses imply that $d$ is at least
three and $p$ is at least five. Let $P$ be a point on $X$ and choose affine 
coordinates $x,y,z$
such that $P=(0,0,0)$ and $T_PX = \{z=0\}$. Near $P$ we can describe $X$ by
$z=u(x,y)$ for some power series $u(x,y)$. At a point $P_0=(x_0,y_0,z_0)$ near
$P$ we can consider lines through $P_0$ given by $L:y-y_0=\l (x-x_0),
z-z_0= \mu (x-x_0)$. Such a line is in $T_{P_0}X$ if and only if
$\mu = u_x(P_0)+u_y(P_0)\l$ and is asymptotic if and only if, in addition,
$u_{xx}(P_0)+2u_{xy}(P_0)\l+u_{yy}(P_0)\l^2 =0$. Also, the line $L$ intersects the
plane at infinity in the point with (affine) coordinates $(\l,\mu)$ and this
gives an analytic expression for the map $\phi$.

We will first show that $\phi$ is ramified everywhere on any two-dimensional
component of $Z_0$ for any surface.
Let $C$ be a one-dimensional component of the locus of planar points, $P$ a
general point of $C$ and $t$ a local parameter for $C$ at $P$ and choose coordinates
$(x,y,z)$ as above. Thus, $u_{xx}=u_{xy}=u_{yy}=0$ along $C$, $(t,\l)$ serve as 
local coordinates for $Z_0$ at $P$ and $\phi$ is given by $(t,\l) \mapsto (\l,u_x+u_y\l)$,
where the partial derivatives are computed at $(x(t),y(t),z(t))$. The derivative of
$\phi$ in these coordinates is given by
$$\pmatrix{\partial \l/ \partial t & \partial \l/\partial \l\cr
\partial (u_x+u_y\l)/ \partial t& \partial (u_x+u_y\l)/ \partial \l}=
\pmatrix{0&1\cr 0&u_y}$$
which has zero determinant, as claimed.

Now we will show that, for an arbitrary surface, the map $\phi$ is ramified at 
a point $(P,L)$ in $Z$ with $P$ a smooth and ordinary point of $X$ if
$L$ is a flecnodal line at $P$. We again choose coordinates $x,y,z$ as above.
Since $P$ is assumed ordinary, there are two asymptotic lines through $P$ on the
plane $z=0$ and we can make a linear change on the coordinates $x,y$ so that these
lines are $x=0$ and $y=0$. We further assume that the flecnodal line $L$ is given
by $y=0$. In terms of $u$ this means that $u=xy+ax^2y+bxy^2+cy^3+\cdots$.
Therefore 
$$\displaylines{
u_{xx} = 2ay +\cdots \cr
u_{xy}= 1 + 2ax +2by +\cdots \cr
u_{yy}= 2bx + 6cy +\cdots.
}
$$
It follows that, if $\l$ and $\mu$ as above give the asymptotic line at points near
$P$ which are close to $L$, then $\l$ and $\mu$ vanish at $P$ and from the above
they have local expansions
$$\displaylines{
\l = -ay +\cdots \cr
\mu = y +\cdots.
}
$$
Thus, The derivative of $\phi$ in the local coordinates $x,y$ is given by
$$\pmatrix{\l_x&\l_y\cr\mu_x&\mu_y}$$
and, at $P=(0,0)$, this gives
$$\pmatrix{0&-a\cr 0&1}$$
and this shows not only that $\phi$ is ramified at $(P,L)$ but also that the
kernel of the derivative of $\phi$ is $L$ regarded as a line in $T_PZ$ which is
isomorphic to $T_PX$ under the projection $Z \to X$.

It follows that, if all points of $X$ are flecnodal and $Z$ is irreducible, 
then $\phi$ is ramified everywhere, so it is inseparable, since
it was proved finite above.
It follows that $p$ divides the degree of $\phi$, namely $d(d-1)(d-2)$, contradicting
$1<d<p$.

{\it Remark:} The conclusion of the theorem still holds, with essentially
the same proof, if
$X$ is defined over some non-algebraically closed subfield of $k$ and $Z$ 
is assumed to be merely irreducible over this subfield. This will be useful
in the sequel.

\proclaim Proposition 1. If $X$ is a split smooth surface of degree $d$ with
$2<d<p$, then not all points are flecnodal. 

{\it Proof:} We proceed as in the proof of the previous theorem.
If all points of $X$ are flecnodal then it follows that $\phi$ is
inseparable on a component $Z_1$, say, of $Z$. Since $Z$ is a double cover
of $X$, we must have $Z_1$ birational to $X$ and we identify (an open set
of) $Z_1$ to (an open set of) $X$ in what follows.
If all points of $X$ are parabolic then $p$ divides $d(d-1)$ which is
impossible for $d<p$. So the generic point of $X$ is ordinary and we
obtain, from the proof of theorem 1, that the derivative of $\phi$ has
rank one at the the generic point.
A section (on an open set) of the kernel of the derivative of $\phi$
gives us a vector field $\d$, which induces a derivation, also denoted
by $\d$ on the function field of $Z_1$.
Therefore, $\phi$ factors through an inseparable map
$Z_1 \to Y$, for some surface $Y$ and this map is induced by the
derivation $\d$. As the derivative of $\phi$ has
rank one at the the generic point, the function field of $Y$
is not contained in the $p$-th powers of the functions on $Z_1$, so we can take
a function $w$ on $Z_1$ (coming from a function on $Y$) which is not 
a $p$-th power
but satisfies $\d w = 0$. The level curves $w = c,c \in k$ of $w$ are tangent to the
vector field corresponding to $\d$. On the other hand, this vector field is contained
in the kernel of the derivative of $\phi$, by construction. Thus, this vector field
is in the direction of the flecnodal lines. This implies that the tangents to
the level curves of $w$ are flecnodal lines.

We now prove a lemma. For the definition of the order sequence
$\e_0,\e_1,\ldots$ of a curve see [SV] section 1.

\proclaim Lemma 1. Assume $p > 3$. If a curve $C$ on a smooth surface $X$ 
is such that the tangent line to $C$ at a generic point is a flecnodal line to 
$X$ at the same point, then either $C$ is contained in the parabolic locus 
of $X$ or the second order of $C \subset \P^3$ satisfies $\e_2 > 2$.

{\it Proof:} We can choose a general point on $C$ and affine coordinates such 
that the point is given by $(0,0,0)$ and, around this point, $X$ is given
by $z=u(x,y)$ for some power series $u$ and $C$ is given by $y=g(x),
z=u(x,g(x))$ for some power series $g$. The condition that the tangent
line to $C$ at a generic point is a flecnodal line to $X$ at the same point
translates into the following conditions, where $\l = g'$:
$$\displaylines{
u_{xx}+2u_{xy}\l+u_{yy}\l^2 =0\cr
u_{xxx}+3u_{xxy}\l+3u_{xyy}\l^2+u_{yyy}\l^3 =0.\cr
}$$

Differentiating the first equation gives 
$$u_{xxx}+3u_{xxy}\l+3u_{xyy}\l^2+u_{yyy}\l^3 + 2(u_{xy}+u_{yy}\l)\l' =0.$$

In view of the second equation, this simplifies to $(u_{xy}+u_{yy}\l)\l' =0$.
So, either $u_{xy}+u_{yy}\l=0$, which means that $\l$ is a double root of 
the equation $u_{xx}+2u_{xy}\l+u_{yy}\l^2 =0$ and thus
$C$ is contained in the parabolic locus of $X$, or we have $g'' = \l' =0$.
If the second option holds, we also have
$u(x,g(x))'' = u_{xx}+2u_{xy}\l+u_{yy}\l^2 + u_y\l' = 0$ proving that
$\e_2> 2$.

Before returning to the proof of the proposition we remark that the above
argument works in characteristic zero and, under this condition, the fact
that $\e_2 > 2$ implies that $C$ is a line. From this it follows the
classical result that a surface in characteristic zero all whose points are
flecnodal is ruled.

Going back to the proof of the proposition, we may assume that not all
level curves of $w$ are parabolic, for otherwise all points of $X$ would
be parabolic and this is impossible for $d<p$. We apply the lemma to a generic
level curve $C$ of $w$. Then, from corollary 1.9
of [SV] we deduce that $\e_2 \ge p$. We proceed to show that 
the intersection multiplicity of $X$ and $T_PC$ at a general point of $C$
is at least $p$.

Using the notation of the lemma, we can adjust the coordinates such that
$g(x) = x^{\e_2} + \cdots$ and $u(x,g(x)) = x^{\e_3} + \cdots$
and this immediately implies that $\partial^j u/\partial x^j(0,0) = 0,
j < \e_2$ which shows that the intersection multiplicity of $X$ and
the line $y=z=0$ is at least $\e_2$, as was to be shown.

Since we assumed that $d<p$ and proved that the intersection multiplicity 
of $X$ and $T_PC$ is at least $p$, we conclude that $T_PC$ is contained in
$X$ for all $P$ in $C$. If $T_PC$ varies with $P$ we conclude that $X$ is
ruled, which is impossible since $X$ is smoooth of degree at least three.
If $T_PC$ is independent of $P$ then $C$ is a line and, as $C$ was a generic
level curve of a non-contant function $w$, again we obtain that $X$ is
ruled, which is impossible. This completes the proof.

\proclaim Corollary 1. If $X$ is a smooth surface in $\P^3$ of degree $d$ with
$2<d<p$, then $X$ contains at most $d(11d-24)$ lines.

{\it Proof:} It is clear that a line contained in $X$ is automatically a
subset of the flecnodal locus. Under the hypotheses of the corollary, the
flecnodal locus is a curve of degree $d(11d-24)$. The result follows.


\bs
{\bf 2. The main result}
\bs

\proclaim Theorem 2.  Let $X$ be a smooth surface in $\P^3$ of degree $d$,
defined over $\F_q$ with $q$ prime and assume that $2<d<q$. 
Let $m$ be the number of lines contained in $X$.
Then
$\# X(\F_q) \le d(d+q-1)(d+2q-2)/6 + m(q+1)$. In particular,
$\# X(\F_q) \le d(d+q-1)(d+2q-2)/6 + d(11d-24)(q+1)$.

{\it Proof:}
We will count the number of points $P$ of $X$ in
an algebraic closure of $\F_q$ whose image under the Frobenius map lies
in one of the asymptotic lines to $X$ at $P$.
This set of points consists of the points $(x_0:\!x_1:\!x_2:\!x_3)$ which satisfy
$f=\sum f_{x_i}x_i^q =
\sum f_{x_i x_j} x_i^qx_j^q =0$.
Consider the scheme $S$ given by the above equations. Clearly $S \subset X$.
By theorem 1 and proposition 1, since we assumed $2<d<q$, not all points 
of $X$ are flecnodal.
We will show that $S$ is not the whole of $X$ and that, outside of
the lines contained in $X$, the rational points of $X$ are isolated 
points of $S$ with multiplicity at least $6$, with equality outside of 
the flecnodal curve.

Since $S$ has degree $d(d+q-1)(d+2q-2)$ and a line has at most $q+1$ rational
points, the result will follow.

Consider then a rational point of $X$. We choose affine coordinates
such that the point is the origin $(0,0,0)$, that the tangent plane is
$z=0$ and that the affine equation for $X$ is $g(x,y,z)=0$. 
We write $g=z+g_2+g_3+\cdots$, where $g_i$ is homogeneous of degree $i$.
Since $q\ge 4$ we get 
that, up to third order terms, the equations cutting down $S$ around $(0,0,0)$
are $g$, 
$$(x^q-x)g_x + (y^q-y)g_y +(z^q-z)g_z = -(xg_x +yg_y +zg_z)+ \cdots=
-z-2g_2-3g_3+ \cdots$$
and
$$\displaylines{-2(d-1)(xg_x +yg_y +zg_z)+x^2g_{xx}+y^2g_{yy}+z^2g_{zz}+
2xyg_{xy}+2xzg_{xz}+2yzg_{yz} + \cdots=\cr
-2(d-1)z-2(2d-3)g_2-3(2d-4)g_3+ \cdots,}$$
which is locally like the intersection of $g_2=0$ and $g_3=0$ on $z=0$ and
therefore intersect at the origin with multiplicity at least six. If
the point is non-flecnodal, $g_2=0$ and $g_3=0$ on $z=0$
have no common factor and therefore intersect at the origin with multiplicity
exactly six.

We will now show that, if a rational point belongs to a curve $C$ inside $S$
then this curve must be a line contained in $X$.
Suppose that the point is the origin of a coordinate system and that
locally, $X$ is given by $z=u(x,y)$
for some power series $u$ and $C$ is given by $y=g(x),
z=u(x,g(x))$ for some power series $g$. 
Now, $S$ is cut out by the equations $z=u(x,y)$,
$u^q-u=u_x(x^q-x)+u_y(y^q-y)$ and 
$u_{xx}(x^q-x)^2+2u_{xy}(x^q-x)(y^q-y)+u_{yy}(y^q-y)^2 =0$.
If we assume that $C$ is contained in $S$ then, differentiating the
second equation with respect to $x$, we get 
$u_{xx}(x^q-x)+u_{xy}((x^q-x)g'+(y^q-y))+u_{yy}(y^q-y)g'=0$, solving for
$g'$ and using the third equation yieds $g'=(y^q-y)/(x^q-x)$. Recalling that
$y=g(x)$ along $C$ this last equation gives $g''=0$.
It also follows then that $u(x,g(x))'' =0$. Now we can conclude that $C$
is a line exactly as in the end of the proof of proposition 1.



{\it Remark:} It is possible to improve the bound in the theorem
somewhat using excess intersection formulae. 


\bs
{\bf 3. Other bounds, consequences and examples}
\bs

In the case of smooth surfaces in $\P^3$ of degree $d$, Deligne's theorem [D]
tells us that, since we have $b_1=b_3=0$ and $b_2=(d^3-4d^2+6d-2)$ for the 
Betti numbers, $\# X(\F_q) \le q^2 + 1+ (d^3-4d^2+6d-2)q$.
An elementary bound, easy to prove is $\# X(\F_q) \le d(q^2 +q+1)$.
This can be improved slightly:

\proclaim Proposition 2. Let $X$ be a smooth surface $\P^3$ of degree $d$,
then $\# X(\F_q) \le (d-1)(q+1)^2 + q+1$. If, moreover $X$ does not contain
a line defined over $\F_q$, then $\# X(\F_q) \le (d-1)q^2 + (d-2)(q+1) +1$.

{\it Proof:}
If $X(\F_q)$ is empty, there is nothing to prove. Otherwise, let
$P \in X(\F_q)$ and consider lines through $P$. Assume first that there
is no line through $P$ defined over $\F_q$ contained in $X$. Of the
$q^2+q+1$ lines through $P$ defined over $\F_q$, those not contained in
$T_PX$ meet $X$ in at most $d-1$ points besides $P$ and those contained in
$T_PX$ meet $X$ in at most $d-2$ points besides $P$. The inequalities  
follow in this case. If $X$ contains a line $L$, consider instead the $q+1$ planes
defined over $\F_q$ containing $L$. Each such plane meets $X$ in a plane
curve of degree $d-1$ besides $L$. This curve has at most $(d-1)(q+1)$ points
defined over $\F_q$ (by an easier version of the same argument). 
Adding up the contribution of all planes together with the $q+1$ points of $L$ 
gives the bound.

Lachaud and Tsfasman have some asymptotic results ([LT],[T]) based on
explicit formulae which do not compare with our results in the range
that our bounds are interesting.

Of course, in our setup $\# X(\F_q) \le \# \P^3(\F_q)= q^3+q^2+q+1$
and this cannot be improved when $d>q$ (see below for an example).
It can be improved however for $d \le q$ as the previous proposition shows.
Our bound only beats these two elementary bounds
for $d < 0.06q$ approximately.
Our bound is better than Deligne's if $d>\sqrt{q/3}$, approximately.

A consequence of our bound is also that, if the number of points of
$X(\F_q)$ is sufficiently large then $X$ must contain lines. One way in
which the number of points of $X(\F_q)$ is large is if many of the
eigenvalues of the Frobenius acting on $H^2(X,\Q_l)$ are equal to $q$.
This is in agreement with the Tate conjecture for surfaces over finite fields. 
This conjecture states
that the rank of the N\'eron-Severi group of a surface $X/\F_q$ coincides
with the multiplicity of $q$ as an eigenvalue of the Frobenius acting on
$H^2(X,\Q_l)$.

Our first example is the surface $x_1^qx_3-x_1x_3^q+x_0^qx_2-x_0x_2^q$
(see [R]), it is smooth of degree $q+1$ and has $q^3+q^2+q+1$ points
over $\F_q$, i.e., it passes through every rational point in $\P^3$. That
shows that one only expects to improve on the bound $q^3+q^2+q+1$
for $d \le q$. Also this surface consists entirely of planar points.

Consider the Fermat surfaces $x_0^d+x_1^d=x_2^d+x_3^d$, over $\F_q$ with
$d|(q-1)$. Then the points with $x_0^d=x_2^d, x_1^d=x_3^d$ give
$(q-1)d^2$ points with $x_0=1$ say. Another $(q-1)d^2$ points come from
$x_0^d=x_3^d, x_1^d=x_2^d$ and a further $(q-1)d^2$ points come from
$x_0^d+x_1^d=x_2^d+x_3^d=0$ if $(q-1)/d$ is even. Some of these points have
been counted twice but the end result is about $2(q-1)d^2$ or $3(q-1)d^2$
according to whether $(q-1)/d$ is odd or even. Note that all these points are
on lines contained in the surface. Sometimes there are more points.
For a numerical example take $d=48$ and $q=1009$. Our bound is
$41879040$, the bound from proposition 2 is $47945710$ and Deligne's bound
is a whopping $103595040$. The actual number of points is a mere $6303792$.
Note that the Hessian of a Fermat surface is $d(d-1)(x_0x_1x_2x_3)^{d-2}$ which 
is a square when $d$ is even, so the surface is split. These surfaces also provide 
examples of pathological geometric behaviour. For instance, for $d=p+1$ all points are 
planar and when $d=p+2$ the generic point is ordinary and flecnodal.

The Schur quartic has $64$ lines and is given by $x_0^4+x_0x_1^3=x_2^4+x_2x_3^3$
and $64$ is the maximum number of lines on a smooth quartic surface in
characteristic zero ([Se]). It might be possible to adapt Segre's proof to
characteristic $p \ge 5$ using the results of section 1.
The Hessian is $81x_0^4x_2^4$ is a square so the surface is split.

The sextic $x_0x_1(x_0^4-x_1^4)=x_2x_3(x_2^4-x_3^4)$ contains $180$ lines
but it is not split. (This example is from [E]).

\bs
{\bf Acknowledgements:} The author would like to thank I. Vainsencher for 
several conversations (fifteen years ago!) on extending [SV] to surfaces.
The author would also like to thank
the NSA (grant MDA904-00-1-0053) for financial support.

\bigskip
\centerline{\bf References.}
\bigskip
\noindent
[D] Deligne, P., {\it La conjecture de Weil I}, Publ. Math. IHES, 43 (1974) 273--307.
\medskip
\noindent
[E] Elkies, N., {\it Curves with many points}, preprint. 
\medskip
\noindent
[HK] Hefez, A. and Kleiman, S., {\it Notes on the duality of projective varieties}. In
Geometry today (Rome, 1984), Progr. Math., 60, Birkh\"auser Boston, 
Boston, MA, 1985. 143--183
\medskip
\noindent
[K] Kiechle, H., {\it Points on Fermat curves over finite fields}, Finite fields: theory, 
applications, and algorithms (Las Vegas, NV, 1993), 181--183, Contemp. Math., 168, 
Amer. Math. Soc., Providence, RI, 1994. 
\medskip
\noindent
[Ku] Kulikov, V. S., {\it Calculus of singularities of imbedding of a general 
algebraic surface in the projective space $\P^3$}. (Russian) Funktsional. 
Anal. i Prilozhen. 17 (1983), no. 3, 15--27. 
\medskip
\noindent
[LT] Lachaud, G, Tsfasman, M. {\it Formules explicites pour le nombre de points 
des vari\'et\'es sur un corps fini}, J. Reine Angew. Math. 493 (1997), 1--60.
\medskip
\noindent
[McS] McCrory, C., Shifrin, T. {\it Cusps of the projective Gauss map}, 
J. Differential Geom. 19 (1984), 257--276. 
\medskip 
\noindent
[R] Rodier, F. {\it Nombre de points des surfaces de Deligne-Lusztig}, 
C. R. Acad. Sci. Paris Ser. I Math. 322 (1996), no. 6, 563--566. 
\medskip
\noindent
[Sa] Salmon, G. {\it A treatise on the analytic geometry of three dimensions} vol. 2,
5th edition, Longmans, Green and Co., London 1915.
\medskip
\noindent
[Sc] Schubert, H. {\it Kalkul der abzhalender geometrie}, Teubner, Leipzig, 1879.
\medskip
\noindent
[Se] Segre, B., {\it The maximum number of lines lying on a quartic surface},
Quart. J. of Math. 14 (1943) 86--96.
\medskip
\noindent
[SV] St\"ohr, K.-O. and Voloch, J. F. {\it Weierstrass points and curves over 
finite fields}, Proc.\ London Math.\ Soc.\ (30) 52(1986), 1-19.
\medskip
\noindent
[T] Tsfasman, M. {\it Nombre de points des surfaces sur un corps fini}, 
Arithmetic, geometry and coding theory (Luminy, 1993), 209--224, de Gruyter, Berlin, 1996.
\medskip
\noindent
[V] Vainsencher, I. {\it Classes caracter\'{\i}sticas em geometria alg\'ebrica},
15$\sp {\rm o}$ Col\'oquio Brasileiro de Matem\'atica.
Instituto de Matem\'atica Pura e Aplicada (IMPA), Rio de Janeiro, 1985. iv+165 pp.
\bigskip
\noindent
Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA
\smallskip
\noindent
e-mail: voloch@math.utexas.edu



\end